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Value at Risk: Methodsintermediate

Linear vs Full-Valuation Methods

VaR engines split by how they revalue a portfolio under market moves. Local (linear, delta) valuation approximates the change in value with first derivatives, so ΔV≈δ ΔS\Delta V \approx \delta\,\Delta S. It is fast and underlies delta-normal VaR but ignores curvature. Full valuation reprices every instrument exactly at each new market scenario, capturing convexity and gamma. Linear methods are fine for portfolios that move proportionally with the market; full valuation is required when options or other nonlinear payoffs dominate.

Why it matters

A linear method draws a straight tangent line through the current price and assumes the portfolio slides along it. For a stock or a forward, that tangent is the truth, and the shortcut is exact. For an option the true payoff bends away from the tangent, so the straight-line guess drifts off the more the market moves. Full valuation throws away the shortcut and reprices each position from scratch in every scenario, paying with computation what it gains in accuracy.

Formulas

Local (delta) approximation
ΔV≈δ ΔS\Delta V \approx \delta\,\Delta S
First-order: the value change is the delta δ\delta times the market move ΔS\Delta S. Exact for linear instruments, approximate once payoffs curve.
Delta-gamma (adds convexity)
ΔV≈δ ΔS+12 γ (ΔS)2\Delta V \approx \delta\,\Delta S + \tfrac{1}{2}\,\gamma\,(\Delta S)^2
A second-order correction using gamma γ\gamma captures curvature for moderate moves, bridging local and full valuation for option books.

Worked examples

Scenario

A trader holds a call option with delta of 0.50 and gamma of 0.04 on a stock at US$100. The stock falls US$10. Compare the delta-only and delta-gamma estimates of the value change.

Solution

Delta-only uses ΔV≈δ ΔS=0.5×(−10)=−5\Delta V \approx \delta\,\Delta S = 0.5 \times (-10) = -5, a loss of US$5 per share. Delta-gamma adds the convexity term, giving ΔV≈0.5(−10)+12(0.04)(−10)2=−3\Delta V \approx 0.5(-10) + \tfrac{1}{2}(0.04)(-10)^2 = -3, a loss of US$3 per share. The large US$10 move makes the convexity term worth US$2, so the linear method overstates the loss. A full-valuation engine would reprice the option exactly and capture this curvature without approximation.

Common mistakes

  • ✗Linear (delta) valuation is accurate for all portfolios. It is exact only for instruments whose value moves proportionally with the market; for options it misses convexity and can badly misstate risk.
  • ✗Full valuation and Monte Carlo are the same thing. Full valuation is a repricing rule; Monte Carlo is a scenario-generation method that can use either local or full repricing.
  • ✗Delta-gamma is as good as full valuation. The second-order term helps for moderate moves but still approximates; large shocks may need exact repricing.
  • ✗Full valuation is always worth the cost. For large linear portfolios the speed of local valuation outweighs the negligible accuracy gain from full repricing.

Revision bullets

  • •Local (delta) valuation: ΔV≈δ ΔS\Delta V \approx \delta\,\Delta S, fast but linear
  • •Full valuation reprices each instrument exactly per scenario
  • •Convexity / gamma matters for option-heavy books
  • •Delta-gamma adds a second-order convexity correction
  • •Choose linear for proportional portfolios, full for nonlinear ones

Quick check

For a portfolio dominated by options, why is full valuation preferred over the delta-normal method?

The delta-gamma approximation improves on delta-only valuation by adding a term proportional to

Connected topics

Parametric VaRHistorical SimMonte Carlo VaRMethod Tradeoffs

Sources

  1. Jorion (2007), Ch. 6
    Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007. ISBN 978-0-07-146495-6.
    Contrasts local (delta, delta-gamma) valuation with full repricing for VaR.
  2. Hull (2018), Ch. 14
    Hull, J. C. Risk Management and Financial Institutions. 5th ed. Wiley, 2018. ISBN 978-1-119-44811-2.
    Discusses the model-building linear approach versus full revaluation for nonlinear portfolios.
How to cite this page
Dr. Phil's Quant Lab. (2026). Linear vs Full-Valuation Methods. Derivatives Atlas. https://phucnguyenvan.com/concept/frm-valuation-method-types
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